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Problem 392

Enmeshed unit circle

A rectilinear grid is an orthogonal grid where the spacing between the gridlines does not have to be equidistant.
An example of such grid is logarithmic graph paper.

Consider rectilinear grids in the Cartesian coordinate system with the following properties:

• The gridlines are parallel to the axes of the Cartesian coordinate system.
• There are N+2 vertical and N+2 horizontal gridlines. Hence there are (N+1) x (N+1) rectangular cells.
• The equations of the two outer vertical gridlines are x = -1 and x = 1.
• The equations of the two outer horizontal gridlines are y = -1 and y = 1.
• The grid cells are colored red if they overlap with the unit circle, black otherwise.

For this problem we would like you to find the postions of the remaining N inner horizontal and N inner vertical gridlines so that the area occupied by the red cells is minimized.

E.g. here is a picture of the solution for N = 10:

The area occupied by the red cells for N = 10 rounded to 10 digits behind the decimal point is 3.3469640797.

Find the positions for N = 400.
Give as your answer the area occupied by the red cells rounded to 10 digits behind the decimal point.

• 网格线均平行于笛卡尔坐标系的坐标轴。
• 一共有N+2条竖直网格线和N+2条水平网格线，因此一共有(N+1) x (N+1) 个长方形网格单元。
• 最外侧两条竖直网格线的方程为x = -1和x = 1。
• 最外侧两条水平网格线的方程为y = -1和y = 1。
• 若网格单元和单位圆重叠则染上红色，否则染上黑色。